Global existence for a damped wave equation and convergence towards a solution of the Navier–Stokes problem

In two and three space dimensions, and under suitable assumptions on the initial data, we show global existence for a damped wave equation which approaches, in some sense, the Navier–Stokes problem. The proofs are based on a refinement of the energy method in Brenier et al. (2004).In this paper, we improve the results of Brenier et al. (2004) and Paicu and Raugel (2007). We relax the regularity of the initial data of the former, even though we still use energy methods as a principal tool. Regarding Paicu and Raugel (2007), the improvement consists in the simplicity of the proofs and in requiring less regularity for the convergence to the Navier–Stokes problem. Indeed, the convergence result we obtain is near-optimal regularity.